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Polarized minimal families of rational curves and higher Fano manifolds
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 134, Number 1, February 2012
- pp. 87-107
- 10.1353/ajm.2012.0008
- Article
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In this paper we investigate Fano manifolds $X$ whose Chern characters ${\rm
ch}_k(X)$ satisfy some positivity conditions. Our approach is via the study
of {\it polarized minimal families of rational curves} $(H_x,L_x)$ through a
general point $x\in X$. First we translate positivity properties of the
Chern characters of $X$ into properties of the pair $(H_x,L_x)$. This allows
us to classify polarized minimal families of rational curves associated to
Fano manifolds $X$ satisfying ${\rm ch}_2(X) \geq 0$ and ${\rm ch}_3(X) \geq
0$. This classification enables us to find new examples of higher Fano
manifolds. We also provide sufficient conditions for these manifolds to be
covered by subvarieties isomorphic to ${\Bbb P}^2$ and ${\Bbb P}^3$.